# Electromagnetic Fields due to Accelerating Charge

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1. Oct 19, 2016

### Yoni V

1. The problem statement, all variables and given/known data
What are the electric and magnetic fields due to a charge that is moving with uniform acceleration?
(Non relativistic)

2. Relevant equations
Retarded solutions for the vector and scalar potentials.

3. The attempt at a solution
My attempt might be an overkill because I'm using the integral solution
$$\psi\left(\boldsymbol{r},t\right)=\int_{V}\frac{\left[f\left(\boldsymbol{r}',t\right)\right]_{ret}}{\left|\boldsymbol{r}-\boldsymbol{r}'\right|}d^{3}r'$$
for the potential as an exercise for correct use of this formula. Perhaps since it's a single particle I can solve for, say, the scalar potential directly from Coulomb's law and just retard the solution. But I'm not sure this is a valid approach because the law essentially assumes electrostatics and immediate information flow without considering the propagation of information through the wave equation (although retarding the solution might just give it the required fix. If anyone can clarify this I'll be grateful). But more importantly, using the above formula I get really mixed up about which of the term is which, and specifically which gets retarded, so I want to be sure I can handle this equation properly for future problems as well.
So I proceed as follows for the scalar potential:

The charge distribution is given by (for simplicity assuming everything is on the x axis):
$$\rho\left(\boldsymbol{r},t\right)=q\delta\left(y\right)\delta\left(z\right)\delta\left(x-\frac{1}{2}at^{2}\right)$$
yielding
$$\phi\left(\boldsymbol{r},t\right)=\int\frac{q\delta\left(y'\right)\delta\left(z'\right)\delta\left(x'-\frac{1}{2}at'^{2}\right)}{\left|\boldsymbol{r}-\boldsymbol{r}'\right|}\delta\left(t'-\left(t-\frac{\left|\boldsymbol{r}-\frac{1}{2}at^{2}\hat{x}\right|}{c}\right)\right)dt'd^{3}r' \\ =\int\frac{q\delta\left(y'\right)\delta\left(z'\right)\delta\left(x'-\frac{1}{2}a\left(t-\frac{\left|\boldsymbol{r}-\frac{1}{2}at^{2}\hat{x}\right|}{c}\right)^{2}\right)}{\left|\boldsymbol{r}-\boldsymbol{r}'\right|}d^{3}r' \\ =\frac{q}{\left|\boldsymbol{r}-\frac{1}{2}a\left(t-\frac{\left|\boldsymbol{r}-\frac{1}{2}at^{2}\hat{x}\right|}{c}\right)^{2}\hat{x}\right|}$$

As can be seen, it really does turn out (at least in this case) just a retardation to the simple solution, but I'm still not sure that I handled the tagging of the variables and retarding them properly.

Is this a correct result? And at which conditions if any may I directly use electrostatics laws such as Coulomb's and retard them in a straightforward fashion? Thanks.

2. Oct 24, 2016